Optimal control of stochastic phase-field models related to tumor growth

Abstract

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

“…requiring that the stochastic basis and the Wiener process are part of the definition of optimal control themselves. This technique mimics the definition of probabilistically weak solution for stochastic evolution equations, and has been employed in other settings such as (Barbu et al 2018;Orrieri et al 2020). In this framework, we prove existence of relaxed optimal controls, and we show that when one restricts the attention only to deterministic controls, then it is possible to get existence in the classical (probabilistically strong) sense.…”

“…In the context of phase-field modelling with stochastic forcing, it is worthwhile mentioning the contributions (Antonopoulou et al 2016;Feireisl and Petcu 2019a, b), as well as (Bauzet et al 2017;Bertacco 2020;Orrieri and Scarpa 2019) on the stochastic Allen-Cahn equation. In the direction of optimal control, we point out (Scarpa 2019b) dealing with a distributed optimal control problem of the stochastic Cahn-Hilliard equation, and the recent work (Orrieri et al 2020) on a stochastic phase-field model for tumour growth.…”

Optimal Distributed Control of a Stochastic Cahn--Hilliard Equation

“…requiring that the stochastic basis and the Wiener process are part of the definition of optimal control themselves. This technique mimics the definition of probabilistically weak solution for stochastic evolution equations, and has been employed in other settings such as [3,64]. In this framework, we prove existence of relaxed optimal controls, and we show that when one restricts the attention only to deterministic controls then it is possible to get existence in the classical (probabilistically strong) sense.…”

Stochastic maximum principle for problems with delay with dependence on the past through general measures

“…The conclusion follows by a standard localization procedure, along the lines given e.g. in [20], end of paragraph 5.4, see also [17], end of the proof of Theorem 5.1, and [24]. 4.…”